Discrete Stochastics

I. Introduction.- 1. Encountering Random.- 2. Specimens of Stochastic Reasoning.- II. Markovian Dynamics.- 1. Finite-state Markovian dynamical systems.- 2. The convex set of stochastic matrices.- 3. The asymptotic behavior of Pn: some special cases.- 4. Asymptotic behavior of P, P2, ...: the method of invariant sets.- III. Discrete Probability Spaces.- 1. The Notion of a Discrete Probability Space (DPS).- 2. Obtaining New Probability Spaces from Given Ones.- 3. Independence.- IV. Independent Identically Distributed (IID) Random Variables.- 1. Addition of independent RVs.- 2. Expectation and Variance.- 3. The Weak Law of Large Numbers (WLLN).- 4. The Central Limit Theorem (CLT) I.- 5. The Central Limit Theorem (CLT) II.- 6. Outlook.- V. Statistics.- 1. Specimens of Statistical Reasoning.- 2. The Game-Theoretical Framework of Statistical Theory.- 3. Tests.- 4. Outlook.- VI. Markov Processes.- 1. Conditional Probabilities.- 2. Markov Processes.- VII. Elements of Information Theory.- 1. Combinatorial and Algebraic Coding Theory.- 2. Source Coding.- 3. Noisy Channels.- VIII. Fluctuation Theory.- 1. The Combinatorial Arcsin Law of Erik Sparre Andersen.- 2. Arcsin.- 3. Symmetrically Distributed Random Variables.- 4. Fluctuations of Random Walks.- 5. The Andersen-Spitzer Formula.- 6. Outlook.- IX. Optimal Strategies in Casinoes: Red and Black.- 1. Strategies and Their Probability of Success.- 2. Some Properties of BOLD.- 3. The Optimality of BOLD for p ? 1/2 ? r.- 4. Non-Optimality of BOLD if p ? 1/2 ? r Fails.- X. Foundational Problems.- 1. The Theory of Randomness.- 2. Subjective Probabilities.- 3. Belief ("bel") Functions.- Appendix A: The Marriage Theorem.- Appendix B: Markovian Semigroups.- Appendix C: One-parameter semigroups of stochastic matrices.